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ParaMetRicEquAtioNs
Extension ONEExtension ONEExtension ONEExtension ONEPreliminary CoursePreliminary CoursePreliminary CoursePreliminary Course
Name _______________
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CAPACITY MATRIX EXTENSION 1TOPIC: Further Parametrics (Ext)
Your say!
What was the most important thing you learned?_________________________
What was something new you learnt? _________________________________
What part(s) of this topic will you need to work on? _____________________
CONTENT CAPACITY BREAKDOWN!DONE
IT!!!!
GOT
IT!!!!!
O
1. Review of the Parabola as a Locus Notes2. Deriving equations of chords, tangents
and normalsNotes
3. Applications of Parametric equations Ex 12.1 (Terry Lee)4. Properties of the parabola Ex 9.2 Q2, 55. Locus problems Ex 12.2 (Terry Lee)
Past HSC questions (Notes)
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The Parabola as a Locus!(A Review)
Put the following phrases into the correct order to complete a true statement for the
Parabola as a Locus.
of a point that is and the fixed line and a fixed lineThe fixed point The locus is called the Focus
is always a Parabola. equidistant from a fixed point is called the Directrix.
SUMMARY!
A parabola is equidistant from a fixed point and a fixed line.
TThe fixed line is called the _____________.
TThe fixed point is called the ______________.
TThe turning point of the parabola is called the ________________.
TThe axis of symmetry of the parabola is called its _______________.
TThe distance between the vertex and the focus is called the _____________________.
TAn interval joining any two points on the parabola is called a _______________.
TA chord that passes through the focus is called a _______________________.
TThe focal chord that is perpendicular to the axis is called the _____________________
TA ________________ is a straight line that touches the parabola at a single point.
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The locus of point P(x, y)moving so that it isequidistant from the point (0, a)and line y = -aisa parabola with equation x2= 4ay
TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to
the focus with length _________
TLatus rectum that is a horizontal focal chordwith length ______
The locus of point P(x, y)moving so that it isequidistant from the point (0, -a)and line y = aisa parabola with equation x2= -4ay
TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to
the focus with length _________TLatus rectum that is a horizontal focal chord
with length ______
The locus of point P(x, y)moving so that it isequidistant from the point (a, 0)and line x = -aisa parabola with equation y2= 4ax
TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to
the focus with length _________
TLatus rectum that is a horizontal focal chordwith length ______
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The locus of point P(x, y)moving so that it isequidistant from the point (-a, 0)and line x = aisa parabola with equation y2= -4ax
TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to
the focus with length _________
TLatus rectum that is a horizontal focal chordwith length ______
The parabola with vertex (h, k)and axis parallelto the y-axis has equation (x h)2= 4a(y k)
TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to
the focus with length _________
TLatus rectum that is a horizontal focal chordwith length ______
The parabola with vertex (h, k)and axis parallelto the x-axis has equation (y k)2= 4a(x h)
TFocus at ________TDirectrix with equation ________TVertex at _______TAxis with equation ________TFocal length the distance from the vertex to
the focus with length _________
TLatus rectum that is a horizontal focal chordwith length ______
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CHORDS, TANGENTS AND NORMALSCHORDS, TANGENTS AND NORMALSCHORDS, TANGENTS AND NORMALSCHORDS, TANGENTS AND NORMALS
e.g., Find the equation of the chord joining points where t=3 and t= -2 on the parabola x =
2at, y = at2
e.g., Find the equation of the tangent to the parabola x2= 8yat the point (4t, 2t2)
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PARAMETRIC FORMULAE
Rehash of previous work:
PROOF
PROOF
If P(2ap, ap) and Q(2aq, aq2) are any twopoints on the parabola x2= 4ay, then the chordPQ has gradient
2
qp +
and equation
( ) 02
1=++ apqxqpy
The parabola x2= 4aycan be
written asx = 2aty = at2
If PQ is a focal chord, then pq= -1
Learn toderive some
of theseequationsrather than
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The tangent to the parabola x2= 4ayat the point P(2ap, ap) has gradientpandequation given by
y px + ap2= 0.
The tangents to the parabola x2= 4ayat points P(2ap, ap) and Q(2aq,aq2) intersect at the point
[a(p + q), apq]
The normal to the curve x2= 4ayat point P(2ap, ap2) has gradient -p
1 and
equation given byx +py = ap3+ 2ap
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The equations of the tangent, normal and chord can also be derived from points in
Cartesian form rather than Parametric form.
The normals to the parabola x2= 4ayat P(2ap, ap2) and Q(2ap, aq2) intersectat
[-apq( p +q ), a( p2+pq + q2+2]
If point A(x1, y1) lies on the parabola ayx 42= , then the equation of the
tangent at A is given byxx1= 2a( y + y1)
If point A(x1, y1) lies on the parabola
= , then the equation of the
normal at A is given by
( )
=
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The equation of the chord of contact XY of tangents drawn from point
(x1, y1) to the parabola ayx 42= is given by
xx1= 2a(y + y1)
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e.g., Tangents are drawn from the point
2
1,
2
1to the points P and Q on the parabola
= . Find the equation of the chord of contact PQ and the co-ordinates of P and Q.
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Eg Find the locus of the midpoints of the chords in the parabola = that pass
through (0, 2).
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PROPERTIES OF A PARABOLAPROVE:
The tangents at the end of a focal chord INTERSECT
AT RIGHT ANGLES on the DIRECTRIX.
PROVE:
The tangent at point P on a parabola is equallyinclined to the axis of the parabola and the focal
chord through P.
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PRACTICE QUESTIONS (APPLICATION)
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SOLUTIONS
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PAST HSC QUESTIONS2009 HSC Q2
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EXT 1 HSC 2008 Q5
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EXT 1 HSC 2007 Q5
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EXT 1 HSC 2006 Q2
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PAST HSC SOLUTIONS2009 HSC Q2
EXT 1 HSC 2008 Q4
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EXT 1 HSC 2007 Q5
EXT 1 HSC 2006 Q2